Singularity Expansion Research Articles

Theoretical aspects of the physical realizability of broad-band equivalent circuits for energy collecting structures

The practical implementation of equivalent circuits that represent the energy delivered to an arbitrary load impedance by an antenna or a port on a scatterer is discussed. Particular attention is paid to the positive-real (PR) function issues associated with the equivalent admittance of the structure when evolved from the point of view of the singularity expansion method (SEM). A proof is given which establishes that the eigenadmittances (reciprocal eigenvalues) of a structure are positive-real functions. The connection between this fact and the positive realness of the admittance associated with individual current eigenmode contributions to total current is discussed. An ad hoc approach to approximating these admittances with simpler pole-pair positive-real admittances is given for high- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> structures. A brief survey of the applicable circuit synthesis algorithms and their features is provided. Some fundamental considerations in the implementation of the sources representing the energy coupled from the incident wave are presented.

The physical content of the singularity expansion method

The singularity expansion method describes the echo return from pulsed radar signals in terms of a Prony-series superposition of damped sinusoids. These are obtained as the residues of poles of the scattering amplitude in the complex frequency plane. For the example of radar scattering from a conducting sphere, we show that these poles can be regrouped into infinite subsets whose residues sums represent individual creeping waves, the nth pole in the subseries corresponding to the resonance caused by a standing creeping wave with n+1/2 wavelength spanning the circumference. The corresponding Prony subseries thus appears simply as a mathematical device which synthesizes the physical creeping wave.

Time-domain techniques in the singularity expansion method

New methods of determining the natural frequencies and natural modes of a structure as characterized in the singularity expansion method (SEM) are presented. The method is based on the time-domain scattering equation which can be east in the form of a matrix difference equation. The homogeneous solution of the difference equation is a series of exponentials as found in the SEM representation. The natural frequency and mode solutions may be obtained either from the determinant of a matrix sum, which is similar to the current frequency-domain search method, or by an eigenvalue approach as in systems theory. The latter has shown promise for efficient SEM computations. An example of each approach is presented.

Unified theory of near-field analysis and measurement: Scattering and inverse scattering

The many scanning procedures of the author's unified theory of near-field analysis and measurement are applied to the determination of complex bistatic scattering patterns of scalar and electromagnetic systems, both with and without correction for the patterns of the probes, one of which may be a compact range. For high accuracy, both the incident and scattered fields are expressed as linear combinations of exact solutions of the differential equations involved (Maxwell's in the electromagnetic cases). For high efficiency, natural orthogonalities with respect to summation are used to decouple the simultaneous equations expressing the measurements in terms of the desired pattern coefficients, implemented by the highly efficient fast Fourier transform as an approximation-free symmetry decomposition. The scanning procedures include spherical, a new, more accurate, more efficient type of plane polar, and many types of plane rectangular, plane radial, and circular cylindrical. All these results are expressed with a single notation and generally applicable equations, based upon symmetry analysis and relativistic invariances. The full apparatus of group representations is applied to reducing the measurement and computational effort, determining symmetries from scattering patterns, and determining Garbacz and singularity expansion method (SEM) modes. Further, Snell's laws, Fresnel's law, and conservation of momenta (e.g., propagation constants) are explained in terms of relativistic invariances.

The K-pulse concept

Two general classes of inversion techniques may be distinguished: those utilizing angular or spatial frequency analysis over as wide a range of target aspects as practicable, and those involving relatively few aspects but employing as wide a frequency spectrum as possible. The latter technique is discussed, and a special excitation waveform called the "kill-pulse" or K-pulse is proposed. By analogy with lumped and distributed networks, a unique aspect-invariant excitation waveform is postulated for an isolated scatterer. This waveform, of finite and minimal duration, then characterizes the pole spectrum of the scatterer as used in the singularity expansion method (SEM) or for target identification. The derivation of this excitation waveform and its relation to one or more surface waves is illustrated by several examples. Using surface waves initiated at certain points of an object, the complex attenuation of circumnavigating waves is estimated by the geometrical theory of diffraction. Three-parameter characteristic equations are derived which predict pole strings with excellent accuracy for conducting spheres and cylinders and lesser accuracy for prolate spheroids of several axial ratios. For spheres and cylinders, the same characteristic equation also yields good estimates of cavity resonance frequencies.

ON THE CONVERGENCE AND NUMERICAL SENSITIVITY OF THE SEM POLE-SERIES IN EARLY-TIME SCATTERING RESPONSE

ABSTRACT The most general form of the Singularity Expansion Method (SEM) representation of the transient scattering response of a finite extent object allows considerable freedom of choice as to the time at which one begins to include the sum of residue contributions into the transient response--i.e., the “turn-on” time. The issue of the chosen form of the coupling coefficient used in computing these residue constituents relates intimately with the choice of turn-on time. The practical range of choices for turn-on time is considerably more restricted than the theoretical one. In this paper limitations on turn-on time are established in terms of the maximum geometric extent of the object and of the geometric extent of the object projected in the direction of propagation of the incident wave. These limitations are dictated in order to insure the convergence of the SEM residue series, and are, in general, different for the Class 1 and for the Class 2 coupling coefficient. It is further argued that the conver...

RADAR ECHO ANALYSIS BY THE SINGULARITY EXPANSION METHOD

ABSTRACT Pulse mode radar operation is analyzed under the assumption that the scattering object Γ lies in the far field of both the transmitter and the receiver. It is shown that, in this approximation, the radar signal is a plane wave s(x · θ0 − t, θ0) near Γ, where θ0 is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e(x θ − t, θ, θ0) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, it is shown that where ŝ(ω,θ0) is the Fourier transform of s(τ,θ0) and T(ωθ,ωθ0) is the scattering amplitude in the direction θ due to the scattering by Γ of a CW mode plane wave with frequency ω and propagation direction θ0. Finally the singularity expansion method is used to show that

MAJOR RESULTS AND UNRESOLVED ISSUES IN SINGULARITY EXPANSION METHOD

ABSTRACT The Singularity Expansion Method (SEM) was derived as a means of interpreting/estimating responses measured during electromagnetic testing of aerospace systems. These responses appeared to consist of a superposition of exponentially damped sinusoidal oscillations. It was shown that the electromagnetic response of a finite-sized, perfectly conducting object to a delta-function incident plane wave is a meromorphic function of the complex frequency. The physical interpretation, computational advantages and fundamental problems associated with using the poles (natural frequencies) of the meromorphic function to construct the transient responses of objects are reviewed. Areas of future investigations, both for the purpose of improving the mathematical foundations and the computational tools are discussed.

THE SINGULARITY EXPANSION METHOD: BACKGROUND AND DEVELOPMENTS

ABSTRACT The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include e...

Impulse response of a conducting sphere based on singularity expansion method

An approximate method based on the singularity expansion method is developed to determine the impulse response of a conducting sphere.

ON THE USE OF SINGULARITY EXPANSION METHOD FOR ANALYSIS OF ANTENNAS IN CONDUCTING MEDIA

ABSTRACT The application of the singularity expansion method (SEM) to the analysis of antennas and electromagnetic scatterers has usually been applied to simple, isolated bodies in free space or to simple bodies near a perfectly conducting ground plane. Theoretical studies of SEM have been applied to these relatively simple geometries to yield significant insight into the radiation and scattering process. In analytically investigating the behavior or antennas in a lossy medium, it is known that in addition to simple pole singularities, there is a branch cut linking two branch points in the complex frequency representation of the antenna response. While significant information regarding the nature of the branch cut and its effect on the antenna response can be obtained by purely analytical methods, a numerical study of this antenna can provide useful results. This paper provides an analysis of the behavior of a linear antenna in a conducting region. Special attention is paid to the importance of the branch...

Theoretical and practical aspects of singularity and eigenmode expansion methods

In recent years a vast amount of literature has been written on singularity and eigenmode expansion methods (SEM and EEM). The main practical purpose is to produce a theory which allows us to identify a flying (or stationary) target from the observed transient field scattered by the target. (The other purposes include representation and computation of the transient field and construction of equivalent circuits according to the object's electromagnetic response.) This problem is discussed. Described are 1) the basic starting points of the engineering approach to the problem and the extent to which they are consistent from the mathematical point of view, 2) the material which has been rigorously established in the field, 3) the important points in practice, and 4) the unsolved mathematical problems in the field. This work may save efforts by other scientists and engineers, since the results demonstrate the kind of research that can be used in practice and the kind which is of mostly theoretical interest.

Calculation of the SEM parameters from the transient response of a thin wire

The problem of determining the singularity expansion method (SEM) parameters from transient thin-wire data is examined. A computer code is used to generate response data. The SEM parameters are computed from these data. For noisy data, it is shown that the parameter values can be improved by averaging.

On the relationship between the singularity expansion method and the mathematical theory of scattering

The so-called natural frequencies in the singularity expansion method (SEM) consist of two nonintersecting sets of wavenumber parameters: namely, a set of interior resonant frequencies which occur on the negative imaginary axis of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> plane and a set of those which reside in the left half of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> plane off the imaginary axis. It is shown that only the latter set can be interpreted as intrinsic to the scatterer. It corresponds to the set of complex poles of the scattering matrix of the problem and also to the set of complex eigenvalues of the related exterior homogeneous boundary value problem. Finally, some doubts about the eigenmode expansion method (EEM) formalism are raised, and a possible justification, based on nonself-adjoint theory, is suggested.

The extraction of the singularity expansion description of a scatterer from sampled transient surface current response

A method is presented whereby one can extract the singularity expansion method (SEM) description of an object's electromagnetic scattering response from spatially sampled transient surface currents. The currents are excited by a known excitation. The SEM data are recoverable to the degree that spatial coupling and (frequency) spectral intensity excite a given SEM mode. Results of a numerical study of the method using the transient response of a thin wire are reported. The data used in the study were computed using a time domain integral equation technique. The ultimate utility of the method lies in the recovery of SEM data from measured data, thus admitting complex-shaped objects into the realm of SEM description. The method is based on a Prony-type pole/residue extraction procedure.

Trajectories of the singularities of a thin wire scatterer parallel to lossy ground

A Pocklington-type integro-differential equation is formulated for the current induced on a cylindrical scatterer near a finitely conducting ground. The Fresnel reflection coefficient for vertical polarization is employed to scale ground-reflected scattered radiation produced by the induced currents. This formulation is reduced to a system of algebraic matrix equations suitable for numerical evaluation through application of the method of moments. The singularity expansion method (SEM) is used to obtain the natural frequencies of the scatterer. Trajectories of these singularities (i.e., natural frequencies) are presented as a function of scatterer height-to-length ratio, ground conductivity, and relative permittivity.

Transient fields of linear antenna arrays

An approximate analytical method is presented, which allows calculation of the transient field strength of linear antenna arrays. The detailed investigation of a logarithmicperiodic dipole antenna on the base of different degrees of approximation shows the separate influence of the element radii and the transient mutual coupling between neighbouring elements. For a better physical understanding of its transient radiation process the cylindrical linear antenna when excited by a step-function with finite rise time is discussed in detail presenting an extension of a recently published approximate version of the Singularity Expansion Method to nonzero center-loading.

Application of Prony's method to time-domain reflectometer data and equivalent circuit synthesis

The singularity expansion of time-domain reflectometer data has been computed and used to calculate lumped-element equivalent circuits for the impedance of some typical antennas. Prony's algorithm was used to obtain the poles of the antennas' terminal voltage waveform due to a step-like excitation. The residues, however, were calculated subject to the uniform-error norm instead of the least-squared-error norm. Although it requires much longer to calculate, the uniform-error norm approximation is often orders of magnitude better than the least-squared-error approximation. The physical realizability of impedance functions obtained from experimental data has been investigated. Also, the procedures necessary to synthesize the lumped-element network have been evaluated. A simple partial-fraction expansion of a realizable impedance function representing an antenna was found to yield nonrealizable element values. A general network synthesis procedure, such as Brune's method, is required.

The singularity expansion method applied to perpendicular crossed wires over a perfectly conducting ground plane

The singularity expansion method (SEM) has been applied to determine natural resonances of a set of perpendicular crossed wires over a perfectly conducting ground plane. The variation of the natural resonances and the mode and coupling vectors have been studied as parameters of the system varied.

Singularity expansions for cylinders of finite conductivity

A new rigorous differential formulation to compute the scattering matrix of an obliquely oriented cylinder of finite conductivity and arbitrary shape is given. The analytic continuation of the scattering operator is examined in the frequency domain as well as in the propagation constant domain. The corresponding computer program is able to compute the singularity expansion for such a cylinder, as well as the propagating mode along it. Numerical examples are given.